Use of Orthogonal Curvilinear Grids for the Representation ...

Use of Orthogonal Curvilinear Grids for the
Representation of the Littoral Ocean Environment

Alan F. Blumberg Richard L. Schaffer
B. Nicholas Kim Matthew T. Calef
Lockheed Martin Information Systems
Sean O’Neil Advanced Simulation Center
HydroQual, Inc.
One Lethbridge Plaza 37 North Ave.
Mahwah, NJ 07430 Burlington, MA 01803
201-529-5151
[email protected] 781-505-9512

James K. Lewis Tom Curtin
Peter J. Stein, Office of Naval Research
Subramaniam D. Rajan
Scientific Solutions, Inc, 800 N. Quincy Dr.
18 Clinton Drive Arlington, VA 22217
Hollis, NH 03049
(603) 880-3784 (703-696-4119)

S. K. Numrich Paul Maassel
Willam Smith
Naval Research Laboratory VisiTech, L.C.
4555 Overlook Ave., SW 3107 N. 18th Str.,
Washington DC, 20375
Arlington, VA 22201

(703-624-1829)

ABSTRACT: The effective representation of the littoral ocean environment has been a long-standing challenge for
the Modeling and Simulation (M&S) community. The need for such a representation continues to grow, as the U.S.
Navy refocuses from deep ocean to littoral missions. One of the greatest challenges of the littoral region is the wide
range of spatial and temporal scales that must be represented. For example, the cross-shore gradients of important
oceanographic parameters are typically an order of magnitude larger than the along-shore gradients. Unfortunately,
the representation techniques typically employed by the real-time M&S community are not well suited to this extreme
range of scales. These representations have included regular Cartesian and geodetic grids, as supported by the IEEE
1278.1a-1998 Gridded Data protocol. This paper describes the application of grids based on an orthogonal
curvilinear system to the littoral environment. Orthogonal curvilinear grids were originally developed by the
computational fluid dynamics community to improve the simulation of fluid flow. Such grids are now routinely used in
numerical models of the littoral ocean environment. They allow a wide range of spatial scales while preserving key
boundaries and maintaining some of the traditional advantages of gridded representations. One of the distinctive
advantages of the use of an orthogonal curvilinear grid systems is the economy of computational and storage
resources that can be attained. Methods for generating operational grids and practical aspects of employing them in
the STOW Synthetic Natural Environment are discussed.

1. Background: which naval forces may engage or otherwise influences
forces or actions on shore. From an oceanographic
With the end of the Cold War, the focus of U. S. Naval perspective, the littoral ocean includes waters on the
strategy has shifted to littoral regions [1]. Littoral continental shelf and in adjacent seas excluding
regions are those "near" a coastline. From a military estuaries. While a good understanding of the deep ocean
perspective, the littoral ocean includes the area from environment was developed during the cold war,

capturing the important phenomenology in the littorals energy can cause greater seasonal fluctuations of
remains a challenge. One of the greatest challenges in temperature due to limited mixing of the deep water
representing the littoral ocean environment for modeling reservoir. Therefore, the less the water depth becomes,
and simulation is the extremely wide range of spatial the greater the water temperature varies due to surface
scales required. This paper describes a representation heating. This surface heating is at its maximum during
that effectively deals with this wide range of spatial the summer season in temperate regions, at which time
scales. This representation is based on orthogonal distinct surface and deep-water separation, called a
curvilinear grids. These grids are applicable both to the thermocline, occurs in the water column. The
generation of littoral ocean data using numerical ocean thermocline limits the mixing of surface and bottom layer
models, and to the use of the output of such models in water. The sound speed in the water column is greatly
military modeling and simulation. affected by the water temperature more than any other
parameters so that sonar operation can be hindered at or
1.1 Physical Basis of the Wide Range of Spatial Scales near the thermocline. Shallow water depth in the littoral
in the Littoral Ocean zone can cause complicated circulation patterns.
Approaching ocean tides, produced mainly by the
The littoral, or coastal, ocean exhibits a great degree of gravitational forces of the moon and sun, interact with
variations of oceanographic parameters compared to the the shallow water depth and the coast. A tidal wave will
deep ocean environment. The most notable examples are change direction and magnitude upon approaching the
the spatial and temporal variations in currents and in the littoral zone due to friction at the bottom of the water
temperature and salinity, and hence density, of seawater. column and the shape of the coastline. Hence the shallow
Factors accounting for these variations are the coast as a water depth often creates much stronger tidal currents
boundary to ocean currents, the shallow water depth over than occur in the deep ocean. Shallow water depth also
the continental shelf, freshwater inflows from rivers and triggers the breaking of wind induced surface waves at
rainfall, and the influence of atmospheric forcing as the shore.
weather moves from continent to coastal waters and vice
versa. Many observations and studies in the littoral zone Freshwater inflow from rivers reduces the salinity of the
confirm the existence of a fundamental length scale, surface layers of water column. With sufficient vertical
which characterizes littoral ocean processes. It is called a mixing, this reduced salinity can penetrate into relatively
baroclinic radius of deformation. Both coastal upwelling deep water. The salinity gradients create horizontal and
and trapped waves are constrained to exist within a few vertical gradients of seawater density near the region
radii of the coast and a few radii also typify cross-front where the freshwater sources enter the littoral zone.
length scales. The baroclinic radius in most of the littoral These horizontal and vertical gradients in turn generate
zone in mid latitudes is of the order of 20 km. density currents, or gravitational currents. The rates of
freshwater inflows can occur with significant seasonal
Naturally, the coast imposes a boundary to ocean variation so that the temporal variation of salinity in the
circulation and hence the ocean currents tend to flow littoral ocean is much greater than that of open ocean.
along the coastline. It is also often found that the currents Riverine inflow may also carry a significant amount of
tend to follow the bathymetry so that the alongshore sediment load into the littoral ocean as well as nutrients.
currents are much stronger than cross-shore currents in Sediment settles to bottom as the current speed decreases
the littoral oceans. In shallow waters, the influx of solar

Figure 1.1. A conceptual illustration of the littoral

in open water, and can cause bottom topographic changes An orthogonal curvilinear grid system preserves right
near the mouth of major rivers. Often fine grained angles between the two coordinates at every point of
sediment travels far into the open water so that the interest in the grid. The region bounded by two adjacent
optical property of sea water changes greatly as compared segments of one of the curvilinear coordinates and two
to deep ocean. Another component controlling sea water adjacent segments of the other curvilinear coordinates
opacity is the micro-organic matter in sea water. Due to will be transformable to a rectangle [3]. A
upwelling of nutrient rich deep water near the straightforward example is the annular region between
continental shelf and the riverine inflows of nutrients, the two concentric circles where one of the curvilinear
littoral zone provides an optimal environment for the coordinates is a radius and the other curvilinear
growth of algae and other micro-organisms. Most of coordinate is made up of a concentric circle placed at a
fisheries in the world ocean are confined in the littoral constant radius in the annular region (Figure 2.1). The
oceans. A conceptual diagram of the processes in the grid is determined by placing several radii at various
littoral ocean is shown in Figure 1.1 [2]. angles around the circle and likewise placing more

2. Orthogonal Curvilinear Grids Figure 2.1. The conformal mapping of annular rings: η denotes
the grid dimension along the circumference of the rings and ξ
The design of a computational grid is a very important
part of modeling. One must choose the proper resolution for the radial direction.
of modeling grid in order to account for spatial variation concentric circles at various radii in the annular region.
of the parameters to be considered. One of the greatest For the above example the mathematical transformation
challenges of the littoral region is the wide range of between the physical grid in the annular region and the
spatial and temporal scales that must be represented. For computational grid represented by a rectangle, may be
example, the cross-shore gradients of important obtained analytically. In general, the mapping
oceanographic parameters are typically an order of coefficients for the irregular boundaries of the littoral
magnitude larger than the along-shore gradients. zone must be computed numerically. Grid generation is
Unfortunately, the representation techniques typically the numerical computation of the mapping coefficients.
employed by the real-time M&S community are not well In practice there are several methods of computing the
suited to this extreme range of scales. These mapping coefficients, but one of the most popular is
representations have included regular Cartesian and conformal, or angle-preserving, mapping. In this method
geodetic grids, as supported by the IEEE 1278.1a-1998 the boundary coordinates are repeatedly subjected to a
Gridded Data protocol. Orthogonal curvilinear grids conformal mapping which maps the each separate
were originally developed by the computational fluid segment to a straight line, except for points at grid
dynamics community to improve the simulation of fluid corners. The points at grid corners are mapped to right
flow. Such grids are now routinely used in numerical angles. Once the boundary points have been distributed, a
models of the littoral ocean environment. They allow a Laplace equation solver can fill in the interior of the
wide range of spatial scales while preserving key domain containing orthogonal grid lines [4].
boundaries and maintaining some of the traditional
advantages of gridded representations. One of the
distinctive advantages of the use of an orthogonal
curvilinear grid systems is the economy of computational
and storage resources that can be attained.

An orthogonal curvilinear coordinate system permits the
design of a grid system in a complicated region such as
that bounded by a shoreline. The transformation between
a complicated and curving grid to a rectangular grid can
be described mathematically. Thus, by incorporating the
grid transformation into the equations of motion
describing the flow physics, the numerical computation
can be thought of as occurring across a rectangular or
Cartesian coordinate, computational grid. The
irregularities of the physical grid are represented by
“mapping coefficients” which describe the
transformation between the physical and the
computational grids.

3. Examples of Application of Orthogonal domain, the resolution increases into the range of 2 km to
Curvilinear Grid System In Littoral Ocean 4 km. The offshore extent of the modeled domain
Modeling: extends out to roughly the 500 m isobath.

An example of orthogonal curvilinear grid applied in
New York Harbor and New York Bight is shown in
Figure 3.1. The modeling domain covers the New York
Bight from Cape May, New Jersey to Nantucket Island
Shoals off the coast of Massachusetts, and most of the
inland water bodies including Block Island Sound, Long
Island Sound, New York Harbor and its tributaries [5].
The domain of the Hudson River extends to the dam at
Troy, New York. The open boundary follows the 100 m
isobath along the continental shelf break. The resolution
of horizontal grid varies from 100 m in the rivers to
about 50 km in the New York Bight. The grid consists of

Figure 3.1. The grid of the New York Harbor region. Figure 3.2. The computational version of the
New York Harbor grid.
49 by 84 segments in the horizontal direction. By
employing the orthogonal curvilinear coordinate grid, the The final example presented here of an orthogonal
model grid has fine resolution in the region of high curvilinear grid is shown in Figure 3.4 [6] and [7]. The
variability and coarse resolution in the areas of less curvilinear grid covers Massachusetts Bay with
variability at the same time. The computational grid horizontal dimension of 68 by 68. The horizontal spacing
representation of the New York Bight model is shown in ranges from 600 m in the vicinity of Boston Harbor to as
Figure 3.2. Notice how the various elements of the large as 6 km along the open boundary in the eastern side
system are connected. of the grid.

An example of orthogonal curvilinear grid for Mamala Figure 3.3. The Oahu Island grid.
Bay, Hawaii is shown in Figure 3.3. A whole island
hydrodynamic model is constructed with increased grid
refinement toward the coast of the Island of Oahu. The
grid consists of 27 by 83 segments in the horizontal
directions. The grid cells in the vicinity of the coast are
of the order of 200 m to 300 m in the cross shelf
direction and, along the oceanic boundaries of the

4. Employing Orthogonal Curvilinear Grids system implicitly defined by the grid itself. This requires

in Military Modeling and Simulation a method of starting with a JointSAF specific coordinate

Figure 3.4. The grid of Massachusetts Bay Figure 4.1. Schematic diagram representing a quad
tree laid on top of a curvilinear grid.
In order to test the utility of orthogonal curvilinear grids
for military modeling and simulation, a representative and locating this point within the curvilinear grid.
application was required. JointSAF is a Computer
Generated Forces (CGF) system developed by the This is a difficult task for the following reason. The
DARPA Synthetic Theater of War (STOW) program[8, curves that make up the curvilinear grid are not the result
9]. JointSAF builds on the U.S. Army's widely used of any mathematical function. Rather they are the result
ModSAF CGF system [10]. JointSAF was selected for of an iterative process described in [3] and [4]. If one
this test because it is widely used, easy to modify, and is were to solve this problem by creating a mathematical
being increasingly employed in Naval exercises and mapping that went from a JointSAF specific coordinate
simulations. These have included Joint Countermine system to a point on a curvilinear grid, one would first
Operational Simulation (JCOS) exercises and recent and have to use a curve fitting algorithm to describe the
upcoming Fleet Battle Experiments. curves mathematically. The resulting function would
then need to be inverted to so that a coordinate in a
4.1 Implementation Approach JointSAF coordinate system would describe a point on
the grid. Given the variance in the grid curvature this
Figure 4.1 represents a quad tree laid on top of a could be a difficult problem to automate. It could also be
curvilinear grid. The quad tree is used to find a good computationally expensive depending on how
starting point for the traversal that will find the four complicated the functions that described the curvilinear
enclosing points in the curvilinear grid. The circles grid turned out to be.
represent the initial starting point and the query point.
The arrows indicate the path of the traversal. In general With this in mind, a high performance lookup, called a
quad tree nodes contain much less data than is depicted quad tree[13], was used. The curvilinear grid is fed into
here. a quad tree, which allows rapid access to spatial data
based on JointSAF specific coordinate systems. The
A significant issue encountered while using a curvilinear quad tree was augmented with a traversal algorithm that
grid is that of getting the desired data quickly. A sensor will locate the four points in the curvilinear grid that
model in JointSAF will request ocean data at a point bound the point being queried.
described in a coordinate system that is native to
JointSAF. The curvilinear grid does not support any The quad tree works in the following manner. First, a
coordinate system, in fact it has it’s own coordinate coordinate system is selected that allows for easy

northerly and westerly comparison of two points. In this All tests were performed on a 550 MHz Pentium III with
case longitude and latitude were used. Next, as JointSAF 512k L2 cache and 256 Mbytes physical memory. As a
starts up, the entire curvilinear grid is read in and point of comparison, access time for a Cartesian grid,
inserted into the quad tree. A quad tree is much like a which can be easily captured in a two-dimensional array,
binary tree, differing in that a node can contain zero to is .36 microseconds.
three data points and that only leaf nodes contain data
points. The top node of the quad tree represents the The results suggest that the fastest lookup is achieved by
entire area covered by the curvilinear grid. The four using the quad tree to find a close point and not the
children of the top node represent the four quadrants closest point. By using a close point and not the closest
created by dividing the area covered by the curvilinear point as the starting point for the traverse of the
grid in half vertically and horizontally each node can curvilinear traverse, the maximum and average number
have children by splitting the region it covers into four of quad tree traverses is greatly reduced, while the
quadrants. A node is split into four sub nodes maximum and average number of curvilinear traverses
(representing four sub quadrants) when a fourth data increases only slightly.
point is added. Quad trees have the benefit that searches
are logarithmic. Finding a point close to the point This approach was integrated into the JointSAF
queried can be achieved quickly by making a single environment architecture, which is based on a system of
descent of the quad tree. Quad trees have a disadvantage models and tags. Tags represent physical attributes such
in that finding the closest point to the point queried is as temperature or current-velocity. Models can then
more expensive. Finding the closest point requires register themselves to provide data for certain tags. The
descending the quad tree multiple times to verify that the environment architecture handles queries for tags. If two
close point is the closest point. models register to provide data for the same tag a
resolver model is used to select the more appropriate
Once the quad tree finds a suitable close point to the model.
point being queried, a traversal of the curvilinear grid is
used to find the four points on the curvilinear grid that We implemented a new model, which provides ocean
enclose the point being queried. information from the littoral zones. The model uses the
lookup described above to locate the correct points on the
This method was tested using the curvilinear grid for curvilinear grid and then access the data. The primary
Onslow Bay. Three separate tests were run. The first purpose of the model is as a proof-of-concept for the
test used the quad tree to find the closest point to the practical use of orthogonal, curvilinear grids in modeling
query point and then begin the traversal of the and simulation. Consequently, we selected the most
curvilinear grid. The second test used the quad tree to expedient method for incorporating orthogonal
get a close point on the curvilinear grid and use that as curvilinear grid data -- predistributed data. The data is
the starting point for the traversal. The third test started provided in a NetCDF data file. NetCDF is a convenient,
from the traversal from the middle of the curvilinear flexible and fast interface to files of the NetCDF format.
grid. The results are shown in table 1. Quad traverse The NetCDF format is designed to support array oriented
indicates the number of times a the algorithm went from scientific data. Since all the locations in the curvilinear
one node in the quad tree to another. Grid traverse grid have been read into memory at start up time, the
indicates the number of times the algorithm went from NetCDF file only needs to be accessed once the four
cell in the curvilinear grid to an adjacent cell. enclosing data points are located. Once the appropriate
values for these data points are located, a 2D or 3D linear
Test 1 Maximum Average Minimum interpolation is performed to get the value at the query
point.
Time (us) 139 26.7 8.70
Quad traverse 333 47.3 17
Grid traverse 4 2.02 1
Test 2 Maximum Average Minimum
Time (us) 13.1 6.50 3.60
Quad traverse 8 7.08 1
Grid traverse 5 2.05 1
Test2 Maximum Average Minimum
Time (us) 174 100 2.00
Quad traverse 00 0
Grid traverse 87 42.7 1

Table 1

Figure 4.2. Integration of orthogonal curvilinear grids in
JSAF.

The implementation within JointSAF is fast, efficient and Author Biographies
provides data at a point from a complex source. The
quad tree and traversal approach can be easily optimized ALAN F. BLUMBERG is Principal Scientist and
to support data queries over a region as opposed to a director of the hydrodynamic group at HydroQual, Inc.
single point. He is the co-developer of the three dimensional
hydrodynamic model, ECOM, which is widely used
We are using a variant of the Princeton Ocean Model as throughout the world.
the source of our data. This ECOM (Estuarine Coastal
Ocean Model) [14] model has recently added the ability B. NICHOLAS KIM is a project manager at
to export data in NetCDF format [15]. Because of the HydroQual, Inc. He has developed and calibrated
immediate availability of orthogonal curvilinear data in hydrodynamic models of many regions since he joined
NetCDF format, this format was employed in the initial HydroQual, Inc. in 1989.
model implementation. It is expected that SEDRIS
interfaces will be addressed at a later stage of the SEAN O'NEIL is an engineer working in the
program. hydrodynamics group at HydroQual, Inc. He is currently
working on various hydrodynamic modeling studies.
The implementation is being conducted in JointSAF 4.9.
Because of the commonality of the underlying RICHARD SCHAFFER is a principal investigator at
environmental libraries, the software should be readily Lockheed Martin Information Systems. He was principle
integratable into similar systems (e.g., ModSAF 5.0 and investigator for the STOW Dynamic Virtual Worlds
JointSAF 5.0). The overall flow of data from the grid program and the ASTT SNE Multi-Resolution Modeling
generation process, through ECOM, and its use by program. Mr. Schaffer has been extensively involved in
JointSAF is illustrated in Figure 4.2. distributed simulation programs and standards since he
joined the SIMNET program in 1985.
4.2 Benefits
MATTHEW T. CALEF is a software engineer at
The greatest benefit of a curvilinear orthogonal grid is Lockheed Martin ASC. His previous work includes
the efficient use of memory and computational resources system engineering on the Naval BFTT program,
to represent a region with highly disparate spatial scales. development of a DDM scheme to help prototype the
The potential magnitude of the benefit can be seen in proposed changes to the HLA specification in draft 1516,
comparing the memory required by a curvilinear grid and fixed wing aircraft behavioral models. Mr. Calef has
with that required by a uniform grid with the same worked extensively with JointSAF since 1996.
maximum resolution. Table 1 provides this comparison
for the Massachusetts Bay, Mamala Bay, and New York JAMES K. LEWIS is a Senior Scientist in
Bight grids described in Section 3. The maximum oceanography at Scientific Solutions, Inc. He has studied
resolutions for each grid system are 0.6km, 0.25km, and coastal and estuarine processes over the past 26 years,
0.1km respectively. The uniform grids are taken to cover has developed and applied a number of models for
the maximum north-south and east-west extents of the various processes, and developed a series of boundary
curvilinear grids. condition formula-tions for ocean models.

Uniform Grid Curvilinear Savings w/ SUBRAMANIAM D. RAJAN is Senior Scientist
Grid Curvilinear Acoustics at Scientific Solutions, Inc. He has worked
extensively on acoustic propagation in shallow water and
Massachusetts 284 x 167 68 x 68 Grid developed an inverse technique to infer bottom properties
Bay 90.25% from field measurements made in the water column.
280 x 360 27 x 83
Mamala Bay 5,000 x 2,400 49 x 84 97.78% S. K. NUMRICH is a Branch Head in Information
99.97% Technology at the Naval Research Laboratory and part
New York time program officer at the Office of Naval Research.
Bight With a background in underwater acoustics, her current
interests lie providing consistent synthetic natural
Table 1: Grid size reduction enabled by curvilinear grids environments for simulation. She serves on a number of
technical panels including international roles in NATO
and TTCP

PETER STEIN is the President and Founder of
Scientific Solutions, Inc. He has worked extensively in
the area of underwater acoustic measurements, modeling,
novel system development.

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